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Galois-theoretic features for 1-smooth pro-p groups

Published online by Cambridge University Press:  29 June 2021

Claudio Quadrelli*
Affiliation:
Department of Mathematics and Applications, University of Milano–Bicocca, MilanoI-20125, Italy
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Abstract

Let p be a prime. A pro-p group G is said to be 1-smooth if it can be endowed with a continuous representation $\theta \colon G\to \mathrm {GL}_1(\mathbb {Z}_p)$ such that every open subgroup H of G, together with the restriction $\theta \vert _H$ , satisfies a formal version of Hilbert 90. We prove that every 1-smooth pro-p group contains a unique maximal closed abelian normal subgroup, in analogy with a result by Engler and Koenigsmann on maximal pro-p Galois groups of fields, and that if a 1-smooth pro-p group is solvable, then it is locally uniformly powerful, in analogy with a result by Ware on maximal pro-p Galois groups of fields. Finally, we ask whether 1-smooth pro-p groups satisfy a “Tits’ alternative.”

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© Canadian Mathematical Society 2021

1 Introduction

Throughout the paper p will denote a prime number, and $\mathbb {K}$ a field containing a root of unity of order p. Let $\mathbb {K}(p)$ denote the compositum of all finite Galois p-extensions of $\mathbb {K}$ . The maximal pro-p Galois group of $\mathbb {K}$ , denoted by $G_{\mathbb {K}}(p)$ , is the Galois group $\operatorname {\mathrm {Gal}}(\mathbb {K}(p)/\mathbb {K})$ , and it coincides with the maximal pro-p quotient of the absolute Galois group of ${\mathbb {K}}$ . Characterising maximal pro-p Galois groups of fields among pro-p groups is one of the most important—and challenging—problems in Galois theory. One of the obstructions for the realization of a pro-p group as maximal pro-p Galois group for some field $\mathbb {K}$ is given by the Artin–Scherier theorem: the only finite group realizable as $G_{\mathbb {K}}(p)$ is the cyclic group of order 2 (cf. [Reference Becker1]).

The proof of the celebrated Bloch-Kato conjecture, completed by Rost and Voevodsky with Weibel’s “patch” (cf. [Reference Haesemeyer and Weibel12Reference Voevodsky27Reference Weibel29]) provided new tools to study absolute Galois groups of field and their maximal pro-p quotients (see, e.g., [Reference Chebolu, Efrat and Mináč2Reference Chebolu, Mináč and Quadrelli3Reference Quadrelli17Reference Quadrelli and Weigel21]). In particular, the now-called Norm Residue Theorem implies that the $\mathbb {Z}/p$ -cohomology algebra of a maximal pro-p Galois group $G_{\mathbb {K}}(p)$

$$ \begin{align*} H^\bullet(G_{\mathbb{K}}(p),\mathbb{Z}/p):=\bigoplus_{n\geq 0} H^n(G_{\mathbb{K}}(p),\mathbb{Z}/p), \end{align*} $$

with $\mathbb {Z}/p$ a trivial $G_{\mathbb {K}}(p)$ -module and endowed with the cup-product, is a quadratic algebra: i.e., all its elements of positive degree are combinations of products of elements of degree 1, and its defining relations are homogeneous relations of degree 2 (see Section 2.3). For instance, from this property one may recover the Artin-Schreier obstruction (see, e.g., [Reference Quadrelli17, Section 2]).

More recently, a formal version of Hilbert 90 for pro-p groups was employed to find further results on the structure of maximal pro-p Galois groups (see [Reference Efrat and Quadrelli9Reference Quadrelli19Reference Quadrelli and Weigel21]). A pair $\mathcal {G}=(G,\theta )$ consisting of a pro-p group G endowed with a continuous representation $\theta \colon G\to \mathrm {GL}_1(\mathbb {Z}_p)$ is called a pro- p pair. For a pro-p pair $\mathcal {G}=(G,\theta )$ let $\mathbb {Z}_p(1)$ denote the continuous left G-module isomorphic to $\mathbb {Z}_p$ as an abelian pro-p group, with G-action induced by $\theta $ (namely, $g.v=\theta (g)\cdot v$ for every $v\in \mathbb {Z}_p(1)$ ). The pair $\mathcal {G}$ is called a Kummerian pro- p pair if the canonical map

$$ \begin{align*} H^1(G,\mathbb{Z}_p(1)/p^n)\longrightarrow H^1(G,\mathbb{Z}_p(1)/p) \end{align*} $$

is surjective for every $n\geq 1$ . Moreover the pair $\mathcal {G}$ is said to be a 1-smooth pro-p pair if every closed subgroup H, endowed with the restriction $\theta \vert _H$ , gives rise to a Kummerian pro-p pair (see Definition 2.1). By Kummer theory, the maximal pro-p Galois group $G_{\mathbb {K}}(p)$ of a field $\mathbb {K}$ , together with the pro-p cyclotomic character $\theta _{\mathbb {K}}\colon G_{\mathbb {K}}(p)\to \mathrm {GL}_1(\mathbb {Z}_p)$ (induced by the action of $G_{\mathbb {K}}(p)$ on the roots of unity of order a p-power lying in $\mathbb {K}(p)$ ) gives rise to a 1-smooth pro-p pair $\mathcal {G}_{\mathbb {K}}$ (see Theorem 2.8).

In [Reference De Clercq and Florence5]—driven by the pursuit of an “explicit” proof of the Bloch–Kato conjecture as an alternative to the proof by Voevodsky—De Clerq and Florence introduced the 1-smoothness property, and formulated the so-called “Smoothness Conjecture”: namely, that it is possible to deduce the surjectivity of the norm residue homomorphism (which is acknowledged to be the “hard part” of the Bloch–Kato conjecture) from the fact that $G_{\mathbb {K}}(p)$ together with the pro-p cyclotomic character is a 1-smooth pro-p pair (see [Reference De Clercq and Florence5, Conjecture 14.25] and [Reference Mináč, Pop, Topaz and Wickelgren15, Section 3.1.6], and Question 2.10).

In view of the Smoothness Conjecture, it is natural to ask which properties of maximal pro-p Galois groups of fields arise also for 1-smooth pro-p pairs. For example, the Artin–Scherier obstruction does: the only finite p-group which may complete into a 1-smooth pro-p pair is the cyclic group $C_2$ of order 2, together with the nontrivial representation $\theta \colon C_2\to \{\pm 1\}\subseteq \mathrm {GL}_1(\mathbb {Z}_2)$ (see Example 2.9).

A pro-p pair $\mathcal {G}=(G,\theta )$ comes endowed with a distinguished closed subgroup: the $\theta $ -center $Z(\mathcal {G})$ of $\mathcal {G}$ , defined by

$$ \begin{align*} Z(\mathcal{G})=\left\langle\: h\in\operatorname{\mathrm{Ker}}(\theta)\:\mid\: ghg^{-1}=h^{\theta(g)}\;\forall\:g\in G\:\right\rangle. \end{align*} $$

This subgroup is abelian, and normal in G. In [Reference Engler and Koenigsmann10], Engler and Koenigsmann showed that if the maximal pro-p Galois group $G_{\mathbb {K}}(p)$ of a field $\mathbb {K}$ is not cyclic then it has a unique maximal normal abelian closed subgroup (i.e., one containing all normal abelian closed subgroups of $G_{\mathbb {K}}(p)$ ), which coincides with the $\theta _{\mathbb {K}}$ -center $Z(\mathcal {G}_{\mathbb {K}})$ , and the short exact sequence of pro-p groups

splits. We prove a group-theoretic analogue of Engler–Koenigsmann’s result for 1-smooth pro-p groups.

Theorem 1.1 Let G be a torsion-free pro-p group, $G\not \simeq \mathbb {Z}_p$ , endowed with a representation $\theta \colon G\to \mathrm {GL}_1(\mathbb {Z}_p)$ such that $\mathcal {G}=(G,\theta )$ is a 1-smooth pro-p pair. Then $Z(\mathcal {G})$ is the unique maximal normal abelian closed subgroup of G, and the quotient $G/Z(\mathcal {G})$ is a torsion-free pro-p group.

In [Reference Ware28], Ware proved the following result on maximal pro-p Galois groups of fields: if $G_{\mathbb {K}}(p)$ is solvable, then it is locally uniformly powerful, i.e., $G_{\mathbb {K}}(p)\simeq A\rtimes \mathbb {Z}_p$ , where A is a free abelian pro-p group, and the right-side factor acts by scalar multiplication by a unit of $\mathbb {Z}_p$ (see Section 3.1). We prove that the same property holds also for 1-smooth pro-p groups.

Theorem 1.2 Let G be a solvable torsion-free pro-p group, endowed with a representation $\theta \colon G\to \mathrm {GL}_1(\mathbb {Z}_p)$ such that $\mathcal {G}=(G,\theta )$ is a 1-smooth pro-p pair. Then G is locally uniformly powerful.

This gives a complete description of solvable torsion-free pro-p groups which may be completed into a 1-smooth pro-p pair. Moreover, Theorem 1.2 settles the Smoothness Conjecture positively for the class of solvable pro-p groups.

Corollary 1.3 If $\mathcal {G}=(G,\theta )$ is a 1-smooth pro-p pair with G solvable, then G is a Bloch–Kato pro-p group, i.e., the $\mathbb {Z}/p$ -cohomology algebra of every closed subgroup of G is quadratic.

Remark 1.4 After the submission of this paper, Snopce and Tanushevski showed in [Reference Snopce and Tanushevski24] that Theorems 1.21.1 hold for a wider class of pro-p groups. A pro-p group is said to be Frattini-injective if distinct finitely generated closed subgroups have distinct Frattini subgroups (cf. [Reference Snopce and Tanushevski24, Definition 1.1]). By [Reference Snopce and Tanushevski24, Theorem 1.11 and Corollary 4.3], a pro-p group which may complete into a 1-smooth pro-p pair is Frattini-injective. By [Reference Snopce and Tanushevski24, Theorem 1.4] a Frattini-injective pro-p group has a unique maximal normal abelian closed subgroup, and by [Reference Snopce and Tanushevski24, Theorem 1.3] a Frattini-injective pro-p group is solvable if, and only if, it is locally uniformly powerful.

A solvable pro-p group does not contain a free nonabelian closed subgroup. For Bloch–Kato pro-p groups—and thus in particular for maximal pro-p Galois groups of fields containing a root of unity of order p—Ware proved the following Tits’ alternative: either such a pro-p group contains a free non-abelian closed subgroup; or it is locally uniformly powerful (see [Reference Ware28, Corollary 1] and [Reference Quadrelli17, Theorem B]). We conjecture that the same phenomenon occurs for 1-smooth pro-p groups.

Conjecture 1.5 Let G be a torsion-free pro-p group which may be endowed with a representation $\theta \colon G\to \mathrm {GL}_1(\mathbb {Z}_p)$ such that $\mathcal {G}=(G,\theta )$ is a 1-smooth pro-p pair. Then either G is locally uniformly powerful, or G contains a closed nonabelian free pro-p group.

2 Cyclotomic pro-p pairs

Henceforth, every subgroup of a pro-p group will be tacitly assumed to be closed, and the generators of a subgroup will be intended in the topological sense.

In particular, for a pro-p group G and a positive integer n, $G^{p^n}$ will denote the closed subgroup of G generated by the $p^n$ th powers of all elements of G. Moreover, for two elements $g,h\in G$ , we set

$$ \begin{align*}h^g=g^{-1}hg,\qquad\text{and}\qquad[h,g]=h^{-1}\cdot h^g,\end{align*} $$

and for two subgroups $H_1,H_2$ of G, $[H_1,H_2]$ will denote the closed subgroup of G generated by all commutators $[h,g]$ with $h\in H_1$ and $g\in H_2$ . In particular, $G'$ will denote the commutator subgroup $[G,G]$ of G, and the Frattini subgroup $G^p\cdot G'$ of G is denoted by $\Phi (G)$ . Finally, $d(G)$ will denote the minimal number of generatord of G, i.e., $d(G)=\dim (G/\Phi (G))$ as a $\mathbb {Z}/p$ -vector space.

2.1 Kummerian pro-p pairs

Let $1+p\mathbb {Z}_p=\{1+p\lambda \mid \lambda \in \mathbb {Z}_p\}\subseteq \mathrm {GL}_1(\mathbb {Z}_p)$ denote the pro-p Sylow subgroup of the group of units of the ring of p-adic integers $\mathbb {Z}_p$ . A pair $\mathcal {G}=(G,\theta )$ consisting of a pro-p group G and a continuous homomorphism

$$ \begin{align*}\theta\colon G\longrightarrow1+p\mathbb{Z}_p\end{align*} $$

is called a cyclotomic pro- p pair, and the morphism $\theta $ is called an orientation of G (cf. [Reference Efrat7, Section 3] and [Reference Quadrelli and Weigel21]).

A cyclotomic pro-p pair $\mathcal {G}=(G,\theta )$ is said to be torsion-free if $\operatorname {\mathrm {Im}}(\theta )$ is torsion-free: this is the case if p is odd; or if $p=2$ and $\operatorname {\mathrm {Im}}(\theta )\subseteq 1+4\mathbb {Z}_2$ . Observe that a cyclotomic pro-p pair $\mathcal {G}=(G,\theta )$ may be torsion-free even if G has nontrivial torsion—e.g., if G is the cyclic group of order p and $\theta $ is constantly equal to 1. Given a cyclotomic pro-p pair $\mathcal {G}=(G,\theta )$ one has the following constructions:

  1. (a) if H is a subgroup of G, $\operatorname {\mathrm {Res}}_H(\mathcal {G})=(H,\theta \vert _H)$ ;

  2. (b) if N is a normal subgroup of G contained in $\operatorname {\mathrm {Ker}}(\theta )$ , then $\theta $ induces an orientation $\bar \theta \colon G/N\to 1+p\mathbb {Z}_p$ , and we set $\mathcal {G}/N=(G/N,\bar \theta )$ ;

  3. (c) if A is an abelian pro-p group, we set $A\rtimes \mathcal {G}=(A\rtimes G,\theta \circ \pi )$ , with $a^g=a^{\theta (g)^{-1}}$ for all $a\in A$ , $g\in G$ , and $\pi $ the canonical projection $A\rtimes G\to G$ .

Given a cyclotomic pro-p pair $\mathcal {G}=(G,\theta )$ , the pro-p group G has two distinguished subgroups:

  1. (a) the subgroup

    (2.1) $$ \begin{align} K(\mathcal{G})=\left\langle\left. h^{-\theta(g)}\cdot h^{g^{-1}}\right|g\in G,h\in\operatorname{\mathrm{Ker}}(\theta)\right\rangle \end{align} $$
    introduced in [Reference Efrat and Quadrelli9, Section 3];
  2. (b) the $\theta $ -center

    (2.2) $$ \begin{align} Z(\mathcal{G})=\left\langle h\in\operatorname{\mathrm{Ker}}(\theta)\left|ghg^{-1}=h^{\theta(g)}\;\forall\:g\in G\right.\right\rangle \end{align} $$
    introduced in [Reference Quadrelli17, Section 1].

Both $Z(\mathcal {G})$ and $K(\mathcal {G})$ are normal subgroups of G, and they are contained in $\operatorname {\mathrm {Ker}}(\theta )$ . Moreover, $Z(\mathcal {G})$ is abelian, while

$$ \begin{align*} K(\mathcal{G})\supseteq\operatorname{\mathrm{Ker}}(\theta)',\qquad\text{and} \qquad K(\mathcal{G})\subseteq\Phi(G). \end{align*} $$

Thus, the quotient $\operatorname {\mathrm {Ker}}(\theta )/K(\mathcal {G})$ is abelian, and if $\mathcal {G}$ is torsion-free one has an isomorphism of pro-p pairs

(2.3) $$ \begin{align} \mathcal{G}/K(\mathcal{G})\simeq(\operatorname{\mathrm{Ker}}(\theta)/K(\mathcal{G}))\rtimes (\mathcal{G}/\operatorname{\mathrm{Ker}}(\theta)), \end{align} $$

namely, $G/K(\mathcal {G})\simeq (\operatorname {\mathrm {Ker}}(\theta )/K(\mathcal {G}))\rtimes (G/\operatorname {\mathrm {Ker}}(\theta ))$ (where the action is induced by $\theta $ , in the latter), and both pro-p groups are endowed with the orientation induced by $\theta $ (cf. [Reference Quadrelli18, Equation 2.6]).

Definition 2.1 Given a cyclotomic pro-p pair $\mathcal {G}=(G,\theta )$ , let $\mathbb {Z}_p(1)$ denote the continuous G-module of rank 1 induced by $\theta $ , i.e., $\mathbb {Z}_p(1)\simeq \mathbb {Z}_p$ as abelian pro-p groups, and $g.\lambda =\theta (g)\cdot \lambda $ for every $\lambda \in \mathbb {Z}_p(1)$ . The pair $\mathcal {G}$ is said to be Kummerian if for every $n\geq 1$ the map

(2.4) $$ \begin{align} H^1(G,\mathbb{Z}_p(1)/p^n)\longrightarrow H^1(G,\mathbb{Z}_p(1)/p), \end{align} $$

induced by the epimorphism of G-modules $\mathbb {Z}_p(1)/p^n\to \mathbb {Z}_p(1)/p$ , is surjective. Moreover, $\mathcal {G}$ is 1-smooth if $\operatorname {\mathrm {Res}}_H(\mathcal {G})$ is Kummerian for every subgroup $H\subseteq G$ .

Observe that the action of G on $\mathbb {Z}_p(1)/p$ is trivial, as $\operatorname {\mathrm {Im}}(\theta )\subseteq 1+p\mathbb {Z}_p$ . We say that a pro-p group G may complete into a Kummerian, or 1-smooth, pro-p pair if there exists an orientation $\theta \colon G\to 1+p\mathbb {Z}_p$ such that the pair $(G,\theta )$ is Kummerian, or 1-smooth.

Kummerian pro-p pairs and 1-smooth pro-p pairs were introduced in [Reference Efrat and Quadrelli9] and in [Reference De Clercq and Florence5, Section 14] respectively. In [Reference Quadrelli and Weigel21], if $\mathcal {G}=(G,\theta )$ is a 1-smooth pro-p pair, the orientation $\theta $ is said to be 1-cyclotomic. Note that in [Reference De Clercq and Florence5, Section 14.1], a pro-p pair is defined to be 1-smooth if the maps (2.4) are surjective for every open subgroup of G, yet by a limit argument this implies also that the maps (2.4) are surjective also for every closed subgroup of G (cf. [Reference Quadrelli and Weigel21, Corollary 3.2]).

Remark 2.1 Let $\mathcal {G}=(G,\theta )$ be a cyclotomic pro-p pair. Then $\mathcal {G}$ is Kummerian if, and only if, the map

$$ \begin{align*} H^1_{\mathrm{cts}}(G,\mathbb{Z}_p(1))\longrightarrow H^1(G,\mathbb{Z}_p(1)/p), \end{align*} $$

induced by the epimorphism of continuous left G-modules $\mathbb {Z}_p(1)\twoheadrightarrow \mathbb {Z}_p(1)/p$ , is surjective (cf. [Reference Quadrelli and Weigel21, Proposition 2.1])—here $H_{\mathrm {cts}}^*$ denotes continuous cochain cohomology as introduced by Tate in [Reference Tate26].

One has the following group-theoretic characterization of Kummerian torsion-free pro-p pairs (cf. [Reference Efrat and Quadrelli9, Theorems 5.6 and 7.1] and [Reference Quadrelli20, Theorem 1.2]).

Proposition 2.2 A torsion-free cyclotomic pro-p pair $\mathcal {G}=(G,\theta )$ is Kummerian if and only if $\operatorname {\mathrm {Ker}}(\theta )/K(\mathcal {G})$ is a free abelian pro-p group.

Remark 2.3 Let $\mathcal {G}=(G,\theta )$ be a cyclotomic pro-p pair with $\theta \equiv \mathbf {1}$ , i.e., $\theta $ is constantly equal to 1. Since $K(\mathcal {G})=G'$ in this case, $\mathcal {G}$ is Kummerian if and only if the quotient $G/G'$ is torsion-free. Hence, by Proposition 2.2, $\mathcal {G}$ is 1-smooth if and only if $H/H'$ is torsion-free for every subgroup $H\subseteq G$ . Pro-p groups with such property are called absolutely torsion-free, and they were introduced by Würfel in [Reference Würfel30]. In particular, if $\mathcal {G}=(G,\theta )$ is a 1-smooth pro-p pair (with $\theta $ nontrivial), then $\operatorname {\mathrm {Res}}_{\operatorname {\mathrm {Ker}}(\theta )}(\mathcal {G})=(\operatorname {\mathrm {Ker}}(\theta ),\mathbf {1})$ is again 1-smooth, and thus $\operatorname {\mathrm {Ker}}(\theta )$ is absolutely torsion-free. Hence, a pro-p group which may complete into a 1-smooth pro-p pair is an absolutely torsion-free-by-cyclic pro-p group.

Example 2.4

  1. (a) A cyclotomic pro-p pair $(G,\theta )$ with G a free pro-p group is 1-smooth for any orientation $\theta \colon G\to 1+p\mathbb {Z}_p$ (cf. [Reference Quadrelli and Weigel21, Section 2.2]).

  2. (b) A cyclotomic pro-p pair $(G,\theta )$ with G an infinite Demushkin pro-p group is 1-smooth if and only if $\theta \colon G\to 1+p\mathbb {Z}_p$ is defined as in [Reference Labute14, Theorem 4] (cf. [Reference Efrat and Quadrelli9, Theorem 7.6]). E.g., if G has a minimal presentation

    $$ \begin{align*} G=\left\langle\:x_1,\ldots,x_d\:\mid\:x_1^{p^f}[x_1,x_2]\cdots[x_{d-1},x_d]=1\:\right\rangle \end{align*} $$
    with $f\geq 1$ (and $f\geq 2$ if $p=2$ ), then $\theta (x_2)=(1-p^f)^{-1}$ , while $\theta (x_i)=1$ for $i\neq 2$ .
  3. (c) For $p\neq 2$ let G be the pro-p group with minimal presentation

    $$ \begin{align*}G=\langle x,y,z\mid [x,y]=z^p\rangle.\end{align*} $$
    Then the pro-p pair $(G,\theta )$ is not Kummerian for any orientation $\theta \colon G\to 1+p\mathbb {Z}_p$ (cf. [Reference Efrat and Quadrelli9, Theorem 8.1]).
  4. (d) Let

    $$ \begin{align*}H=\left\{\left(\begin{array}{ccc} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 &1 \end{array}\right)\mid a,b,c\in\mathbb{Z}_p\right\}\end{align*} $$
    be the Heisenberg pro-p group. The pair $(H,\mathbf {1})$ is Kummerian, as $H/H'\simeq \mathbb {Z}_p^2$ , but H is not absolutely torsion-free. In particular, H can not complete into a 1-smooth pro-p pair (cf. [Reference Quadrelli18, Example 5.4]).
  5. (e) The only 1-smooth pro-p pair $(G,\theta )$ with G a finite p-group is the cyclic group of order 2 $G\simeq \mathbb {Z}/2$ , endowed with the only nontrivial orientation $\theta \colon G\twoheadrightarrow \{\pm 1\}\subseteq 1+2\mathbb {Z}_2$ (cf. [Reference Efrat and Quadrelli9, Example 3.5]).

Remark 2.5 By Example 2.4(e), if $\mathcal {G}=(G,\theta )$ is a torsion-free 1-smooth pro-p pair, then G is torsion-free.

A torsion-free pro-p pair $\mathcal {G}=(G,\theta )$ is said to be $\theta $ -abelian if the following equivalent conditions hold:

  1. (i) $\operatorname {\mathrm {Ker}}(\theta )$ is a free abelian pro-p group, and $\mathcal {G}\simeq \operatorname {\mathrm {Ker}}(\theta )\rtimes (\mathcal {G}/\operatorname {\mathrm {Ker}}(\theta ))$ ;

  2. (ii) $Z(\mathcal {G})$ is a free abelian pro-p group, and $Z(\mathcal {G})=\operatorname {\mathrm {Ker}}(\theta )$ ;

  3. (iii) $\mathcal {G}$ is Kummerian and $K(\mathcal {G})=\{1\}$

(cf. [Reference Quadrelli17, Proposition 3.4] and [Reference Quadrelli20, Section 2.3]). Explicitly, a torsion-free pro-p pair $\mathcal {G}=(G,\theta )$ is $\theta $ -abelian if and only if G has a minimal presentation

(2.5) $$ \begin{align} G=\left\langle x_0,x_i,i\in I\mid [x_0,x_i]=x_i^{q},[x_i,x_j]=1\:\forall\:i,j\in I\right\rangle\simeq\mathbb{Z}_p^I\rtimes \mathbb{Z}_p \end{align} $$

for some set I and some p-power q (possibly $q=p^\infty =0$ ), and in this case $\operatorname {\mathrm {Im}}(\theta )=1+q\mathbb {Z}_p$ . In particular, a $\theta $ -abelian pro-p pair is also 1-smooth, as every open subgroup U of G is again isomorphic to $\mathbb {Z}_p^I\rtimes \mathbb {Z}_p$ , with action induced by $\theta \vert _U$ , and therefore $\operatorname {\mathrm {Res}}_U(\mathcal {G})$ is $\theta \vert _U$ -abelian.

Remark 2.6 From [Reference Efrat and Quadrelli9, Theorem 5.6], one may deduce also the following group-theoretic characterization of Kummerian pro-p pairs: a pro-p group G may complete into a Kummerian oriented pro-p group if, and only if, there exists an epimorphism of pro-p groups $\varphi \colon G\twoheadrightarrow \bar G$ such that $\bar G$ has a minimal presentation (2.5), and $\operatorname {\mathrm {Ker}}(\varphi )$ is contained in the Frattini subgroup of G (cf., e.g., [Reference Quadrelli and Weigel22, Proposition 3.11]).

Remark 2.7 If $G\simeq \mathbb {Z}_p$ , then the pair $(G,\theta )$ is $\theta $ -abelian, and thus also 1-smooth, for any orientation $\theta \colon G\to 1+p\mathbb {Z}_p$ .

On the other hand, if $\mathcal {G}=(G,\theta )$ is a $\theta $ -abelian pro-p pair with $d(G)\geq 2$ , then $\theta $ is the only orientation which may complete G into a 1-smooth pro-p pair. Indeed, let $\mathcal {G}'=(G,\theta ')$ be a cyclotomic pro-p pair, with $\theta '\colon G\to 1+p\mathbb {Z}_p$ different to $\theta $ , and let $\{x_0,x_i,i\in I\}$ be a minimal generating set of G as in the presentation (2.5)—thus, $\theta (x_i)=1$ for all $i\in I$ , and $\theta (x_0)\in 1+q\mathbb {Z}_p$ . Then for some $i\in I$ one has $\theta '\vert _H\not \equiv \theta \vert _H$ , with H the subgroup of G generated by the two elements $x_0$ and $x_i$ . In particular, one has $\theta ([x_0,x_i])=\theta '([x_0,x_i])=1$ .

Suppose that $\mathcal {G}'$ is 1-smooth. If $\theta '(x_i)\neq 1$ , then

$$ \begin{align*} x_i^q=x_i\cdot x_i^q\cdot x_i^{-1}=(x_i^q)^{\theta'(x_i)}=x_i^{q\theta'(x_i)}, \end{align*} $$

hence $x_i^{q(1-\theta '(x_i))}=1$ , a contradiction as G is torsion-free by Remark 2.5. If $\theta '(x_i)=1$ then necessarily $\theta '(x_0)\neq \theta (x_0)$ , and thus

$$ \begin{align*} x_i^{\theta(x_0)}=x_0\cdot x_i\cdot x_0^{-1}=x_i^{\theta'(x_0)}, \end{align*} $$

hence $x_i^{\theta (x_0)-\theta '(x_0)}=1$ , again a contradiction as G is torsion-free. (See also [Reference Quadrelli and Weigel21, Corollary 3.4].)

2.2 The Galois case

Let $\mathbb {K}$ be a field containing a root of 1 of order p, and let $\mu _{p^\infty }$ denote the group of roots of 1 of order a p-power contained in the separable closure of $\mathbb {K}$ . Then $\mu _{p^\infty }\subseteq \mathbb {K}(p)$ , and the action of the maximal pro-p Galois group $G_{\mathbb {K}}(p)=\operatorname {\mathrm {Gal}}(\mathbb {K}(p)/\mathbb {K})$ on $\mu _{p^\infty }$ induces a continuous homomorphism

$$ \begin{align*}\theta_{\mathbb{K}}\colon G_{\mathbb{K}}(p)\longrightarrow1+p\mathbb{Z}_p\end{align*} $$

—called the pro- p cyclotomic character of $G_{\mathbb {K}}(p)$ —as the group of the automorphisms of $\mu _{p^{\infty }}$ which fix the roots of order p is isomorphic to $1+p\mathbb {Z}_p$ (see, e.g., [Reference Efrat8, p. 202] and [Reference Efrat and Quadrelli9, Section 4]). In particular, if $\mathbb {K}$ contains a root of 1 of order $p^k$ for $k\geq 1$ , then $\operatorname {\mathrm {Im}}(\theta _{\mathbb {K}})\subseteq 1+p^k\mathbb {Z}_p$ .

Set $\mathcal {G}_{\mathbb {K}}=(G_{\mathbb {K}}(p),\theta _{\mathbb {K}})$ . Then by Kummer theory one has the following (see, e.g., [Reference Efrat and Quadrelli9, Theorem 4.2]).

Theorem 2.8 Let $\mathbb {K}$ be a field containing a root of 1 of order p. Then $\mathcal {G}_{\mathbb {K}}=(G_{\mathbb {K}}(p),\theta _{\mathbb {K}})$ is 1-smooth.

1-smooth pro-p pairs share the following properties with maximal pro-p Galois groups of fields.

Example 2.9

  1. (a) The only finite p-group which occurs as maximal pro-p Galois group for some field $\mathbb {K}$ is the cyclic group of order 2, and this follows from the pro-p version of the Artin–Schreier Theorem (cf. [Reference Becker1]). Likewise, the only finite p-group which may complete into a 1-smooth pro-p pair, is the cyclic group of order 2 (endowed with the only nontrivial orientation onto $\{\pm 1\}$ ), as it follows from Example 2.4(e) and Remark 2.5.

  2. (b) If x is an element of $G_{\mathbb {K}}(2)$ for some field $\mathbb {K}$ and x has order 2, then x self-centralizes (cf. [Reference Craven and Smith4, Proposition 2.3]). Likewise, if x is an element of a pro- $2$ group G which may complete into a 1-smooth pro-2 pair, then x self-centralizes (cf. [Reference Quadrelli and Weigel21, Section 6.1]).

2.3 Bloch–Kato and the Smoothness Conjecture

A non-negatively graded algebra $A_\bullet =\bigoplus _{n\geq 0}A_n$ over a field $\mathbb {F}$ , with $A_0=\mathbb {F}$ , is called a quadratic algebra if it is one-generated—i.e., every element is a combination of products of elements of degree 1—and its relations are generated by homogeneous relations of degree 2. One has the following definitions (cf. [Reference De Clercq and Florence5, Definition 14.21] and [Reference Quadrelli17, Section 1]).

Definition 2.2 Let G be a pro-p group, and let $n\geq 1$ . Cohomology classes in the image of the natural cup-product

$$ \begin{align*} H^1(G,\mathbb{Z}/p)\times\ldots\times H^1(G,\mathbb{Z}/p)\overset{\cup}{\longrightarrow} H^n(G,\mathbb{Z}/p) \end{align*} $$

are called symbols (relative to $\mathbb {Z}/p$ , wieved as trivial G-module).

  1. (i) If for every open subgroup $U\subseteq G$ every element $\alpha \in H^n(U,\mathbb {Z}/p)$ , for every $n\geq 1$ , can be written as

    $$ \begin{align*}\alpha=\mathrm{cor}_{V_1,U}^n(\alpha_1)+\cdots+\mathrm{cor}^n_{V_r,U}(\alpha_r),\end{align*} $$
    with $r\geq 1$ , where $\alpha _i\in H^n(V_i,\mathbb {Z}/p)$ is a symbol and
    $$ \begin{align*}\mathrm{cor}_{V_i,U}^n\colon H^n(V_i,\mathbb{Z}/p)\longrightarrow H^n(U,\mathbb{Z}/p)\end{align*} $$
    is the corestriction map (cf. [Reference Neukirch, Schmidt and Wingberg16, Chapter I, Section 5]), for some open subgroups $V_i\subseteq U$ , then G is called a weakly Bloch–Kato pro- p group.
  2. (ii) If for every closed subgroup $H\subseteq G$ the $\mathbb {Z}/p$ -cohomology algebra

    $$ \begin{align*}H^\bullet(H,\mathbb{Z}/p)=\bigoplus_{n\geq0}H^n(H,\mathbb{Z}/p),\end{align*} $$
    endowed with the cup-product, is a quadratic algebra over $\mathbb {Z}/p$ , then G is called a Bloch–Kato pro- p group. As the name suggests, a Bloch–Kato pro-p group is also weakly Bloch-Kato.

By the Norm Residue Theorem, if $\mathbb {K}$ contains a root of unity of order p, then the maximal pro-p Galois group $G_{\mathbb {K}}(p)$ is Bloch–Kato. The pro-p version of the “Smoothness Conjecture,” formulated by De Clerq and Florence, states that being 1-smooth is a sufficient condition for a pro-p group to be weakly Bloch–Kato (cf. [Reference De Clercq and Florence5, Conjugation 14.25]).

Conjecture 2.10 Let $\mathcal {G}=(G,\theta )$ be a 1-smooth pro-p pair. Then G is weakly Bloch–Kato.

In the case of $\mathcal {G}=\mathcal {G}_{\mathbb {K}}$ for some field $\mathbb {K}$ containing a root of 1 of order p, using Milnor K-theory one may show that the weak Bloch–Kato condition implies that $H^\bullet (G,\mathbb {Z}/p)$ is one-generated (cf. [Reference De Clercq and Florence5, Rem. 14.26]). In view of Theorem 2.8, a positive answer to the Smoothness Conjecture would provide a new proof of the surjectivity of the norm residue isomorphism, i.e., the “surjectivity” half of the Bloch–Kato conjecture (cf. [Reference De Clercq and Florence5, Section 1.1]).

Conjecture 2.10 has been settled positively for the following classes of pro-p groups.

  1. (a) Finite p-groups: indeed, if $\mathcal {G}=(G,\theta )$ is a 1-smooth pro-p pair with G a finite (nontrivial) p-group, then by Example 2.4–(e) $p=2$ , G is a cyclic group of order two and $\theta \colon G\twoheadrightarrow \{\pm 1\}$ , so that $\mathcal {G}\simeq (\operatorname {\mathrm {Gal}}(\mathbb {C}/\mathbb {R}),\theta _{\mathbb {R}})$ , and G is Bloch–Kato.

  2. (b) Analytic pro-p groups: indeed if $\mathcal {G}=(G,\theta )$ is a 1-smooth pro-p pair with G a p-adic analytic pro-p group, then by [Reference Quadrelli18, Theorem 1.1] G is locally uniformly powerful and thus Bloch–Kato (see § 3.1 below).

  3. (c) Pro-p completions of right-angled Artin groups: indeed, in [Reference Snopce and Zalesskii25], it is shown that if $\mathcal {G}=(G,\theta )$ is a 1-smooth pro-p pair with G the pro-p completion of a right-angled Artin group induced by a simplicial graph $\Gamma $ , then necessarily $\theta $ is trivial and $\Gamma $ has the diagonal property—namely, G may be constructed starting from free pro-p groups by iterating the following two operations: free pro-p products, and direct products with $\mathbb {Z}_p$ —and thus G is Bloch–Kato (cf. [Reference Snopce and Zalesskii25, Theorem 1.2]).

3 Normal abelian subgroups

3.1 Powerful pro-p groups

Definition 3.1 A finitely generated pro-p group G is said to be powerful if one has $G'\subseteq G^p$ , and also $G'\subseteq G^4$ if $p=2$ . A powerful pro-p group which is also torsion-free and finitely generated is called a uniformly powerful pro-p group.

For the properties of powerful and uniformly powerful pro-p groups, we refer to [Reference Dixon, du Sautoy, Mann and Segal6, Chapter 4].

A pro-p group whose finitely generated subgroups are uniformly powerful, is said to be locally uniformly powerful. As mentioned in Section 1, a pro-p group G is locally uniformly powerful if, and only if, G has a minimal presentation (2.5)—i.e., G is locally powerful if, and only if, there exists an orientation $\theta \colon G\to 1+p\mathbb {Z}_p$ such that $(G,\theta )$ is a torsion-free $\theta $ -abelian pro-p pair (cf. [Reference Quadrelli17, Theorem A] and [Reference Chebolu, Mináč and Quadrelli3, Proposition 3.5]).

Therefore, a locally uniformly powerful pro-p group G comes endowed automatically with an orientation $\theta \colon G\to 1+p\mathbb {Z}_p$ such that $\mathcal {G}=(G,\theta )$ is a 1-smooth pro-p pair. In fact, finitely generated locally uniformly powerful pro-p groups are precisely those uniformly powerful pro-p groups which may complete into a 1-smooth pro-p pair (cf. [Reference Quadrelli18, Proposition 4.3]).

Proposition 3.1 Let $\mathcal {G}=(G,\theta )$ be a 1-smooth torsion-free pro-p pair. If G is locally powerful, then $\mathcal {G}$ is $\theta $ -abelian, and thus G is locally uniformly powerful.

It is well-known that the $\mathbb {Z}/p$ -cohomology algebra of a pro-p group G with minimal presentation (2.5) is the exterior $\mathbb {Z}/p$ -algebra

$$ \begin{align*} H^\bullet(H,\mathbb{Z}/p)\simeq \bigwedge_{n\geq0} H^1(H,\mathbb{Z}/p) \end{align*} $$

—if $p=2$ then $\bigwedge _{n\geq 0} V$ is defined to be the quotient of the tensor algebra over $\mathbb {Z}/p$ generated by V by the two-sided ideal generated by the elements $v\otimes v$ , $v\in V$ —so that $H^\bullet (G,\mathbb {Z}/p)$ is quadratic. Moreover, every subgroup $H\subseteq G$ is again locally uniformly powerful, and thus also $H^\bullet (H,\mathbb {Z}/p)$ is quadratic. Hence, a locally uniformly powerful pro-p group is Bloch–Kato.

3.2 Normal abelian subgroups of maximal pro-p Galois groups

Let $\mathbb {K}$ be a field containing a root of 1 of order p (and also $\sqrt {-1}$ if $p=2$ ). In Galois theory, one has the following result, due to Engler et al. (cf. [Reference Engler and Nogueira11] and [Reference Engler and Koenigsmann10]).

Theorem 3.2 Let $\mathbb {K}$ be a field containing a root of 1 of order p (and also $\sqrt {-1}$ if $p=2$ ), and suppose that the maximal pro-p Galois group $G_{\mathbb {K}}(p)$ of $\mathbb {K}$ is not isomorphic to $\mathbb {Z}_p$ . Then $G_{\mathbb {K}}(p)$ contains a unique maximal abelian normal subgroup.

By [Reference Quadrelli and Weigel21, Theorem 7.7], such a maximal abelian normal subgroup coincides with the $\theta _{\mathbb {K}}$ -center $Z(\mathcal {G}_{\mathbb {K}})$ of the pro-p pair $\mathcal {G}_{\mathbb {K}}=(G_{\mathbb {K}}(p),\theta _{\mathbb {K}})$ induced by the pro-p cyclotomic character $\theta _{\mathbb {K}}$ (cf. § 2.2). Moreover, the field $\mathbb {K}$ admits a p-Henselian valuation with residue characteristic not p and non-p-divisible value group, such that the residue field $\kappa $ of such a valuation gives rise to the cyclotomic pro-p pair $\mathcal {G}_{\kappa }$ isomorphic to $\mathcal {G}_{\mathbb {K}}/Z(\mathcal {G}_{\mathbb {K}})$ , and the induced short exact sequence of pro-p groups

(3.1)

splits (cf. [Reference Engler and Koenigsmann10, Section 1] and [Reference Efrat8, Example 22.1.6]—for the definitions related to p-henselian valuations of fields, we direct the reader to [Reference Efrat8, Section 15.3]). In particular, $G_{\mathbb {K}}(p)/Z(\mathcal {G}_{\mathbb {K}})$ is torsion-free.

Remark 3.3 By [Reference Quadrelli and Weigel21, Theorems 1.2 and 7.7], Theorem 3.2 and the splitting of (3.1) generalize to 1-smooth pro-p pairs whose underlying pro-p group is Bloch–Kato. Namely, if $\mathcal {G}=(G,\theta )$ is a 1-smooth pro-p pair with G a Bloch–Kato pro-p group, then $Z(\mathcal {G})$ is the unique maximal abelian normal subgroup of G, and it has a complement in G.

3.3 Proof of Theorem 1.1

In order to prove Theorem 1.1 (and also Theorem 1.2 later on), we need the following result.

Proposition 3.4 Let $\mathcal {G}=(G,\theta )$ be a torsion-free 1-smooth pro-p pair, with $d(G)=2$ and $G=\langle x,y\rangle $ . If $[[x,y],y]=1$ , then $\operatorname {\mathrm {Ker}}(\theta )=\langle y\rangle $ and

$$ \begin{align*}xyx^{-1}=y^{\theta(x)}.\end{align*} $$

Proof Let H be the subgroup of G generated by y and $[x,y]$ . Recall that by Remark 2.5, G (and hence also H) is torsion-free.

If $d(H)=1$ then $H\simeq \mathbb {Z}_p$ , as H is torsion-free. Moreover, H is generated by y and $x^{-1}yx$ , and thus $xHx^{-1}\subseteq H$ . Therefore, x acts on $H\simeq \mathbb {Z}_p$ by multiplication by $1+p\lambda $ for some $\lambda \in \mathbb {Z}_p$ . If $\lambda =0$ then G is abelian, and thus $G\simeq \mathbb {Z}_p^2$ as it is absolutely torsion-free, and $\theta \equiv \mathbf {1}$ by Remark 2.7. If $\lambda \neq 0$ then x acts nontrivially on the elements of H, and thus $\langle x\rangle \cap H=\{1\}$ and $G=H\rtimes \langle x\rangle $ : by (2.5), $(G,\theta ')$ is a $\theta '$ -abelian pro-p pair, with $\theta '\colon G\to 1+p\mathbb {Z}_p$ defined by $\theta '(x)=1+p\lambda $ and $\theta '(y)=1$ . By Remark 2.7, one has $\theta '\equiv \theta $ , and thus $\theta (x)=1+p\lambda $ and $\theta (y)=1$ .

If $d(H)=2$ , then H is abelian by hypothesis, and torsion-free, and thus $(H,\theta ')$ is $\theta '$ -abelian, with $\theta '\equiv {\mathbf {1}}\colon H\to 1+p\mathbb {Z}_p$ trivial. By Remark 2.7, one has $\theta '=\theta \vert _H$ , and thus $y,[x,y]\in \operatorname {\mathrm {Ker}}(\theta )$ . Now put $z=[x,y]$ and $t=y^p$ , and let U be the open subgroup of G generated by $x,z,t$ . Clearly, $\operatorname {\mathrm {Res}}_U(\mathcal {G})$ is again 1-smooth. By hypothesis one has $z^y=z$ , and hence commutator calculus yields

(3.2) $$ \begin{align} [x,t]=[x,y^p]=z\cdot z^y\cdots z^{y^{p-1}}=z^p. \end{align} $$

Put $\lambda =1-\theta (x)^{-1}\in p\mathbb {Z}_p$ . Since $t\in \operatorname {\mathrm {Ker}}(\theta )$ , by (2.1) $[x,t]\cdot t^{-\lambda }$ lies in $K(\operatorname {\mathrm {Res}}_U(\mathcal {G}))$ . Since t and z commute, from (3.2) one deduces

(3.3) $$ \begin{align} [x,t]t^{-\lambda}=z^pt^{-\lambda}=z^pt^{-\frac{\lambda}{p}p}=\left(zt^{-\lambda/p}\right)^p\in K(\operatorname{\mathrm{Res}}_U(\mathcal{G})). \end{align} $$

Moreover, $zt^{-\lambda /p}\in \operatorname {\mathrm {Ker}}(\theta \vert _U)$ . Since $\operatorname {\mathrm {Res}}_U(\mathcal {G})$ is 1-smooth, by Proposition 2.2, the quotient $\operatorname {\mathrm {Ker}}(\theta \vert _U)/K(\operatorname {\mathrm {Res}}_U(\mathcal {G}))$ is a free abelian pro-p group, and therefore (3.3) implies that also $zt^{-\lambda /p}$ is an element of $K(\operatorname {\mathrm {Res}}_U(\mathcal {G}))$ .

Since $K(\operatorname {\mathrm {Res}}_U(\mathcal {G}))\subseteq \Phi (U)$ , one has $z\equiv t^{\lambda /p}\bmod \Phi (U)$ . Then by [Reference Dixon, du Sautoy, Mann and Segal6, Proposition 1.9] $d(U)=2$ and U is generated by x and t. Since $[x,t]\in U^p$ by (3.2), the pro-p group U is powerful. Therefore, $\operatorname {\mathrm {Res}}_U(\mathcal {G})$ is $\theta \vert _U$ -abelian by Proposition 3.1. In particular, the subgroup $K(\operatorname {\mathrm {Res}}_U(\mathcal {G}))$ is trivial, and thus

$$ \begin{align*}[x,y]=z=t^{\lambda/p}=y^{1-\theta(x)^{-1}},\end{align*} $$

and the claim follows.▪

Proposition 3.4 is a generalization of [Reference Quadrelli18, Proposition 5.6].

Theorem 3.5 Let $\mathcal {G}=(G,\theta )$ be a torsion-free 1-smooth pro-p pair, with $d(G)\geq 2$ .

  1. (i) The $\theta $ -center $Z(\mathcal {G})$ is the unique maximal abelian normal subgroup of G.

  2. (ii) The quotient $G/Z(\mathcal {G})$ is a torsion-free pro-p group.

Proof Recall that G is torsion-free by Remark 2.5. Since $Z(\mathcal {G})$ is an abelian normal subgroup of G by definition, in order to prove (i) we need to show that if A is an abelian normal subgroup of G, then $A\subseteq Z(\mathcal {G})$ .

First, we show that $A\subseteq \operatorname {\mathrm {Ker}}(\theta )$ . If $A\simeq \mathbb {Z}_p$ , let y be a generator of A. For every $x\in G$ one has $xyx^{-1}\in A$ , and thus $xyx^{-1}=y^{\lambda }$ , for some $\lambda \in 1+p\mathbb {Z}_p$ . Let H be the subgroup of G generated by x and y, for some $x\in G$ such that $d(H)=2$ . Then the pair $(H,\theta ')$ is $\theta '$ -abelian for some orientation $\theta '\colon H\to 1+p\mathbb {Z}_p$ such that $y\in \operatorname {\mathrm {Ker}}(\theta ')$ , as H has a presentation as in (2.5). Since both $\operatorname {\mathrm {Res}}_H(\mathcal {G})$ and $(H,\theta ')$ are 1-smooth pro-p pairs, by Remark 2.7, one has $\theta '=\theta \vert _H$ , and thus $A\subseteq \operatorname {\mathrm {Ker}}(\theta )$ .

If $A\not \simeq \mathbb {Z}_p$ , then A is a free abelian pro-p group with $d(A)\geq 2$ , as G is torsion-free. Therefore, by Remark 2.3 the pro-p pair $(A,\mathbf {1})$ is 1-smooth. Since also $\operatorname {\mathrm {Res}}_A(\mathcal {G})$ is 1-smooth, Remark 2.7 implies that $\theta \vert _A=\mathbf {1}$ , and hence $A\subseteq \operatorname {\mathrm {Ker}}(\theta )$ .

Now, for arbitrary elements $x\in G$ and $y\in A$ , put $z=[x,y]$ . Since A is normal in G, one has $z\in A$ , and since A is abelian, one has $[z,y]=1$ . Then Proposition 3.4 applied to the subgroup of G generated by $\{x,y\}$ yields $xyx^{-1}=x^{\theta (x)}$ , and this completes the proof of statement (i).

In order to prove statement (ii), suppose that $y^p\in Z(\mathcal {G})$ for some $y\in G$ . Then $y^p\in \operatorname {\mathrm {Ker}}(\theta )$ , and since $\operatorname {\mathrm {Im}}(\theta )$ has no nontrivial torsion, also y lies in $\operatorname {\mathrm {Ker}}(\theta )$ . Since G is torsion-free by Remark 2.5, $y^p\neq 1$ . Let H be the subgroup of G generated by y and x, for some $x\in G$ such that $d(H)\geq 2$ . Since $xy^px^{-1}=(y^p)^{\theta (x)}$ , commutator calculus yields

(3.4) $$ \begin{align} y^{p(1-\theta(x)^{-1})}=[x,y^p]=[x,y]\cdot[x,y]^y\cdots[x,y]^{y^{p-1}}. \end{align} $$

Put $z=[x,y]$ , and let S be the subgroup of H generated by $y,z$ . Clearly, $\operatorname {\mathrm {Res}}_S(\mathcal {G})$ is 1-smooth, and since $y,z\in \operatorname {\mathrm {Ker}}(\theta )$ , one has $\theta \vert _S=\mathbf {1}$ , and thus $S/S'$ is a free abelian pro-p group by Remark 2.3. From (3.4) one deduces

(3.5) $$ \begin{align} y^{p(1-\theta(x)^{-1})}\cdot z^{-p}\equiv \left(y^{1-\theta(x)^{-1}}\cdot z^{-1}\right)^p\equiv 1\ \mod S'. \end{align} $$

Since $S/S'$ is torsion-free, (3.5) implies that $z\equiv y^{1-\theta (x)^{-1}}\bmod \Phi (S)$ , so that S is generated by y, and $S\simeq \mathbb {Z}_p$ , as G is torsion-free. Therefore, $S'=\{1\}$ , and (3.5) yields $[x,y]=y^{1-\theta (x)^{-1}}$ , and this completes the proof of statement (ii).▪

Remark 3.6 Let G be a pro-p group isomorphic to $\mathbb {Z}_p$ , and let $\theta \colon G\to 1+p\mathbb {Z}_p$ be a nontrivial orientation. Then by Example 2.4(a), $\mathcal {G}=(G,\theta )$ is 1-smooth. Since G is abelian and $\theta (x)\neq 1$ for every $x\in G$ , $x\neq 1$ , $Z(\mathcal {G})=\{1\}$ , still every subgroup of G is normal and abelian.

In view of the splitting of (3.1) (and in view of Remark 3.3), it seems natural to ask the following question.

Question 3.7 Let $\mathcal {G}=(G,\theta )$ be a torsion-free 1-smooth pro-p pair, with $d(G)\geq 2$ . Is the pro-p pair $\mathcal {G}/Z(\mathcal {G})=(G/Z(\mathcal {G}),\bar {\theta })$ 1-smooth? Does the short exact sequence of pro-p groups

split?

If $\mathcal {G}=(G,\theta )$ is a torsion-free pro-p pair, then either $\operatorname {\mathrm {Ker}}(\theta )=G$ , or $\operatorname {\mathrm {Im}}(\theta )\simeq \mathbb {Z}_p$ , hence in the former case one has $G\simeq \operatorname {\mathrm {Ker}}(\theta )\rtimes (G/\operatorname {\mathrm {Ker}}(\theta ))$ , as the right-side factor is isomorphic to $\mathbb {Z}_p$ , and thus p-projective (cf. [Reference Neukirch, Schmidt and Wingberg16, Chapter III, Section 5]). Since $Z(\mathcal {G})\subseteq Z(\operatorname {\mathrm {Ker}}(\theta ))$ (and $Z(\mathcal {G})= Z(G)$ if $\operatorname {\mathrm {Ker}}(\theta )=G$ ), and since $\operatorname {\mathrm {Ker}}(\theta )$ is absolutely torsion-free if $\mathcal {G}$ is 1-smooth, Question 3.7 is equivalent to the following question (of its own group-theoretic interest): if G is an absolutely torsion-free pro-p group, does G split as direct product

$$ \begin{align*}G\simeq Z(G)\times (G/Z(G))\:\text{?}\end{align*} $$

One has the following partial answer (cf. [Reference Würfel30, Proposition 5]): if G is absolutely torsion-free, and $Z(G)$ is finitely generated, then $\Phi _n(G)=Z(\Phi _n(G))\times H$ , for some $n\geq 1 $ and some subgroup $H\subseteq \Phi _n(G)$ (here $\Phi _n(G)$ denotes the iterated Frattini series of G, i.e., $\Phi _1(G)=G$ and $\Phi _{n+1}(G)=\Phi (\Phi _n(G))$ for $n\geq 1$ ).

4 Solvable pro-p groups

4.1 Solvable pro-p groups and maximal pro-p Galois groups

Recall that a (pro-p) group G is said to be meta-abelian if there is a short exact sequence

such that both N and $\bar G$ are abelian; or, equivalently, if the commutator subgroup $G'$ is abelian. Moreover, a pro-p group G is solvable if the derived series $(G^{(n)})_{n\geq 1}$ of G—i.e., $G^{(1)}=G$ and $G^{(n+1)}=[G^{(n)},G^{(n)}]$ —is finite, namely $G^{(N+1)}=\{1\}$ for some finite N.

Example 4.1 A nonabelian locally uniformly powerful pro-p group G is meta-abelian: if $\theta \colon G\to 1+p\mathbb {Z}_p$ is the associated orientation, then $G'\subseteq \operatorname {\mathrm {Ker}}(\theta )^p$ , and thus $G'$ is abelian.

In Galois theory, one has the following result by Ware (cf. [Reference Ware28, Theorem 3], see also [Reference Koenigsmann13] and [Reference Quadrelli17, Theorem 4.6]).

Theorem 4.2 Let $\mathbb {K}$ be a field containing a root of 1 of order p (and also $\sqrt {-1}$ if $p=2$ ). If the maximal pro-p Galois group $G_{\mathbb {K}}(p)$ is solvable, then $\mathcal {G}_{\mathbb {K}}$ is $\theta _{\mathbb {K}}$ -abelian.

4.2 Proof of Theorem 1.2 and Corollary 1.3

In order to prove Theorem 1.2, we prove first the following intermediate results—a consequence of Würfel’s result [Reference Würfel30, Proposition 2] —, which may be seen as the “1-smooth analogue” of [Reference Ware28, Theorem 2].

Proposition 4.3 Let $\mathcal {G}=(G,\theta )$ be a torsion-free 1-smooth pro-p pair. If G is meta-abelian, then $\mathcal {G}$ is $\theta $ -abelian.

Proof Assume first that $\theta \equiv \mathbf {1}$ —i.e., G is absolutely torsion-free (cf. Remark 2.3). Then G is a free abelian pro-p group by [Reference Würfel30, Proposition 2].

Assume now that $\theta \not \equiv \mathbf {1}$ . Since $\mathcal {G}$ is 1-smooth, also $\operatorname {\mathrm {Res}}_{\operatorname {\mathrm {Ker}}(\theta )}(\mathcal {G})$ and $\operatorname {\mathrm {Res}}_{\operatorname {\mathrm {Ker}}(\theta )'}(\mathcal {G})$ are 1-smooth pro-p pairs, and thus $\operatorname {\mathrm {Ker}}(\theta )$ and $\operatorname {\mathrm {Ker}}(\theta )'$ are absolutely torsion-free. Moreover, $\operatorname {\mathrm {Ker}}(\theta )'\subseteq G'$ , and since the latter is abelian, also $\operatorname {\mathrm {Ker}}(\theta )'$ is abelian, i.e., $\operatorname {\mathrm {Ker}}(\theta )$ is meta-abelian. Thus $\operatorname {\mathrm {Ker}}(\theta )$ is a free abelian pro-p group by [Reference Würfel30, Proposition 2]. Consequently, for arbitrary $y\in \operatorname {\mathrm {Ker}}(\theta )$ and $x\in G$ , the commutator $[x,y]$ lies in $\operatorname {\mathrm {Ker}}(\theta )$ and $[[x,y],y]=1$ . Therefore, Proposition 3.4 implies that $xyx^{-1}=y^{\theta (y)}$ for every $x\in G$ and $y\in \operatorname {\mathrm {Ker}}(\theta )$ , namely, $\mathcal {G}$ is $\theta $ -abelian.▪

Note that Proposition 4.3 generalizes [Reference Würfel30, Proposition 2] from absolutely torsion-free pro-p groups to 1-smooth pro-p groups. From Proposition 4.3, we may deduce Theorem 1.2.

Proposition 4.4 Let $\mathcal {G}=(G,\theta )$ be a torsion-free 1-smooth pro-p pair. If G is solvable, then G is locally uniformly powerful.

Proof Let N be the positive integer such that $G^{(N)}\neq \{1\}$ and $G^{(N+1)}=\{1\}$ . Then for every $1\leq n\leq N$ , the pro-p pair $\operatorname {\mathrm {Res}}_{G^{n}}(\mathcal {G})$ is 1-smooth, and $G^{(n)}$ is solvable, and moreover $\theta \vert _{G^{(n)}}\equiv \mathbf {1}$ if $n\geq 2$ .

Suppose that $N\geq 3$ . Since $G^{(N-1)}$ is meta-abelian and $\theta \vert _{G^{(N-1)}}\equiv \mathbf {1}$ , Proposition 4.3 implies that $G^{(N-1)}$ is a free abelian pro-p group, and therefore $G^{(N)}=\{1\}$ , a contradiction. Thus, $N\leq 2$ , and G is meta-abelian. Therefore, Proposition 4.3 implies that the pro-p pair $\mathcal {G}$ is $\theta $ -abelian, and hence G is locally uniformly powerful (cf. § 3.1).▪

Proposition 4.4 may be seen as the 1-smooth analogue of Ware’s Theorem 4.2. Corollary 1.3 follows from Proposition 4.4 and from the fact that a locally uniformly powerful pro-p group is Bloch–Kato (cf. § 3.1).

Corollary 4.5 Let $\mathcal {G}=(G,\theta )$ be a torsion-free 1-smooth pro-p pair. If G is solvable, then G is Bloch–Kato.

This settles the Smoothness Conjecture for the class of solvable pro-p groups.

4.3 A Tits’ alternative for 1-smooth pro-p groups

For maximal pro-p Galois groups of fields one has the following Tits’ alternative (cf. [Reference Ware28, Corollary 1]).

Theorem 4.6 Let $\mathbb {K}$ be a field containing a root of 1 of order p (and also $\sqrt {-1}$ if $p=2$ ). Then either $\mathcal {G}_{\mathbb {K}}$ is $\theta _{\mathbb {K}}$ -abelian, or $G_{\mathbb {K}}(p)$ contains a closed nonabelian free pro-p group.

Actually, the above Tits’ alternative holds also for the class of Bloch–Kato pro-p groups, with p odd: if a Bloch–Kato pro-p group G does not contain any free nonabelian subgroups, then it can complete into a $\theta $ -abelian pro-p pair $\mathcal {G}=(G,\theta )$ (cf. [Reference Quadrelli17, Theorem B], this Tits’ alternative holds also for $p=2$ under the further assumption that the Bockstein morphism $\beta \colon H^1(G,\mathbb {Z}/2)\to H^2(G,\mathbb {Z}/2)$ is trivial, see [Reference Quadrelli17, Theorem 4.11]).

Clearly, a solvable pro-p group contains no free nonabelian subgroups.

A pro-p group is p-adic analytic if it is a p-adic analytic manifold and the map $(x,y)\mapsto x^{-1} y$ is analytic, or, equivalently, if it contains an open uniformly powerful subgroup (cf. [Reference Dixon, du Sautoy, Mann and Segal6, Theorem 8.32])—e.g., the Heisenberg pro-p group is analytic. Similarly to solvable pro-p groups, a p-adic analytic pro-p group does not contain a free nonabelian subgroup (cf. [Reference Dixon, du Sautoy, Mann and Segal6, Corollary 8.34]).

Even if there are several p-adic analytic pro-p groups which are solvable (e.g., finitely generated locally uniformly powerful pro-p groups), none of these two classes of pro-p groups contains the other one: e.g.,

  1. (a) the wreath product $\mathbb {Z}_p\wr \mathbb {Z}_p\simeq \mathbb {Z}_p^{\mathbb {Z}_p}\rtimes \mathbb {Z}_p$ is a meta-abelian pro-p group, but it is not p-adic analytic (cf. [Reference Shalev23]) and

  2. (b) if G is a pro-p-Sylow subgroup of $\mathrm {SL}_2(\mathbb {Z}_p)$ , then G is a p-adic analytic pro-p group, but it is not solvable.

In addition, it is well-known that also for the class of pro-p completions of right-angled Artin pro-p groups one has a Tits’ alternative: the pro-p completion of a right-angled Artin pro-p group contains a free nonabelian subgroup unless it is a free abelian pro-p group (i.e., unless the associated graph is complete)—and thus it is locally uniformly powerful.

In [Reference Quadrelli18], it is shown that analytic pro-p groups which may complete into a 1-smooth pro-p pair are locally uniformly powerful. Therefore, after the results in [Reference Quadrelli18] and [Reference Snopce and Zalesskii25], and Theorem 1.2, it is natural to ask whether a Tits’ alternative, analogous to Theorem 4.6 (and its generalization to Bloch–Kato pro-p groups), holds also for all torsion-free 1-smooth pro-p pairs.

Question 4.7 Let $\mathcal {G}=(G,\theta )$ be a torsion-free 1-smooth pro-p pair, and suppose that $\mathcal {G}$ is not $\theta $ -abelian. Does G contain a closed nonabelian free pro-p group?

In other words, we are asking whether there exists torsion-free 1-smooth pro-p pairs $\mathcal {G}=(G,\theta )$ such that G is not analytic nor solvable, and yet it contains no free nonabelian subgroups. In view of Theorem 4.6 and of the Tits’ alternative for Bloch–Kato pro-p groups [Reference Quadrelli17, Theorem B], a positive answer to Question 4.7 would corroborate the Smoothness Conjecture.

Observe that—analogously to Question 3.7—Question 4.7 is equivalent to asking whether an absolutely torsion-free pro-p group which is not abelian contains a closed nonabelian free subgroup. Indeed, by Proposition 3.4 (in fact, just by [Reference Quadrelli18, Proposition 5.6]), if $\mathcal {G}=(G,\theta )$ is a torsion-free 1-smooth pro-p pair and $\operatorname {\mathrm {Ker}}(\theta )$ is abelian, then $\mathcal {G}$ is $\theta $ -abelian.

Acknowledgment

The author thanks I. Efrat, J. Minac, N.D. Tân, and Th. Weigel for working together on maximal pro-p Galois groups and their cohomology; and P. Guillot and I. Snopce for the interesting discussions on 1-smooth pro-p groups. Also, the author wishes to thank the editors of CMB-BMC, for their helpfulness, and the anonymous referee.

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